Abstract The Seidel energy of a simple graph G is the sum of the absolute values of the eigenvalues of the Seidel matrix of G. In this paper we study the Seidel eigenvalues of complete multipartite graphs and find the exact value of the Seidel energy of the complete multipartite graphs.
{"title":"Seidel energy of complete multipartite graphs","authors":"M. Oboudi","doi":"10.1515/spma-2020-0131","DOIUrl":"https://doi.org/10.1515/spma-2020-0131","url":null,"abstract":"Abstract The Seidel energy of a simple graph G is the sum of the absolute values of the eigenvalues of the Seidel matrix of G. In this paper we study the Seidel eigenvalues of complete multipartite graphs and find the exact value of the Seidel energy of the complete multipartite graphs.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"212 - 216"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0131","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67302124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Bolla, T. Szabados, Máté Baranyi, Fatma Abdelkhalek
Abstract Given a weakly stationary, multivariate time series with absolutely summable autocovariances, asymptotic relation is proved between the eigenvalues of the block Toeplitz matrix of the first n autocovariances and the union of spectra of the spectral density matrices at the n Fourier frequencies, as n → ∞. For the proof, eigenvalues and eigenvectors of block circulant matrices are used. The proved theorem has important consequences as for the analogies between the time and frequency domain calculations. In particular, the complex principal components are used for low-rank approximation of the process; whereas, the block Cholesky decomposition of the block Toeplitz matrix gives rise to dimension reduction within the innovation subspaces. The results are illustrated on a financial time series.
{"title":"Block circulant matrices and the spectra of multivariate stationary sequences","authors":"M. Bolla, T. Szabados, Máté Baranyi, Fatma Abdelkhalek","doi":"10.1515/spma-2020-0121","DOIUrl":"https://doi.org/10.1515/spma-2020-0121","url":null,"abstract":"Abstract Given a weakly stationary, multivariate time series with absolutely summable autocovariances, asymptotic relation is proved between the eigenvalues of the block Toeplitz matrix of the first n autocovariances and the union of spectra of the spectral density matrices at the n Fourier frequencies, as n → ∞. For the proof, eigenvalues and eigenvectors of block circulant matrices are used. The proved theorem has important consequences as for the analogies between the time and frequency domain calculations. In particular, the complex principal components are used for low-rank approximation of the process; whereas, the block Cholesky decomposition of the block Toeplitz matrix gives rise to dimension reduction within the innovation subspaces. The results are illustrated on a financial time series.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"36 - 51"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0121","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42596897","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Symmetric matrix classes of bandwidth 2r + 1 was studied in 1972 through binomial coefficients. In this paper, non-symmetric matrix classes with the binomial coefficients are considered where r + s + 1 is the bandwidth, r is the lower bandwidth and s is the upper bandwidth. Main results for inverse, determinants and norm-infinity of inverse are presented. The binomial coefficients are used for the derivation of results.
{"title":"A class of symmetric and non-symmetric band matrices via binomial coefficients","authors":"Omojola Micheal, E. Kılıç","doi":"10.1515/spma-2020-0142","DOIUrl":"https://doi.org/10.1515/spma-2020-0142","url":null,"abstract":"Abstract Symmetric matrix classes of bandwidth 2r + 1 was studied in 1972 through binomial coefficients. In this paper, non-symmetric matrix classes with the binomial coefficients are considered where r + s + 1 is the bandwidth, r is the lower bandwidth and s is the upper bandwidth. Main results for inverse, determinants and norm-infinity of inverse are presented. The binomial coefficients are used for the derivation of results.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"321 - 330"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0142","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47329418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Following the recent work of Zheng et al., in this paper, we first present a new extension Hartfiel’s determinant inequality to multiple positive definite matrices, and then we extend the result to a larger class of matrices, namely, matrices whose numerical ranges are contained in a sector. Our result complements that of Mao.
{"title":"Further extensions of Hartfiel’s determinant inequality to multiple matrices","authors":"Wenhui Luo","doi":"10.1515/spma-2020-0125","DOIUrl":"https://doi.org/10.1515/spma-2020-0125","url":null,"abstract":"Abstract Following the recent work of Zheng et al., in this paper, we first present a new extension Hartfiel’s determinant inequality to multiple positive definite matrices, and then we extend the result to a larger class of matrices, namely, matrices whose numerical ranges are contained in a sector. Our result complements that of Mao.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"78 - 82"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0125","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47472126","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Given a square matrix A, replacing each of its nonzero entries with the symbol * gives its zero-nonzero pattern. Such a pattern is said to be spectrally arbitrary when it carries essentially no information about the eigenvalues of A. A longstanding open question concerns the smallest possible number of nonzero entries in an n × n spectrally arbitrary pattern. The Generalized 2n Conjecture states that, for a pattern that meets an appropriate irreducibility condition, this number is 2n. An example of Shitov shows that this irreducibility is essential; following his technique, we construct a smaller such example. We then develop an appropriate algebraic condition and apply it computationally to show that, for n ≤ 7, the conjecture does hold for ℝ, and that there are essentially only two possible counterexamples over ℂ. Examining these two patterns, we highlight the problem of determining whether or not either is in fact spectrally arbitrary over ℂ. A general method for making this determination for a pattern remains a major goal; we introduce an algebraic tool that may be helpful.
{"title":"Algebraic conditions and the sparsity of spectrally arbitrary patterns","authors":"Louis Deaett, C. Garnett","doi":"10.1515/spma-2020-0136","DOIUrl":"https://doi.org/10.1515/spma-2020-0136","url":null,"abstract":"Abstract Given a square matrix A, replacing each of its nonzero entries with the symbol * gives its zero-nonzero pattern. Such a pattern is said to be spectrally arbitrary when it carries essentially no information about the eigenvalues of A. A longstanding open question concerns the smallest possible number of nonzero entries in an n × n spectrally arbitrary pattern. The Generalized 2n Conjecture states that, for a pattern that meets an appropriate irreducibility condition, this number is 2n. An example of Shitov shows that this irreducibility is essential; following his technique, we construct a smaller such example. We then develop an appropriate algebraic condition and apply it computationally to show that, for n ≤ 7, the conjecture does hold for ℝ, and that there are essentially only two possible counterexamples over ℂ. Examining these two patterns, we highlight the problem of determining whether or not either is in fact spectrally arbitrary over ℂ. A general method for making this determination for a pattern remains a major goal; we introduce an algebraic tool that may be helpful.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"257 - 274"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0136","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48658253","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A list of complex numbers Λ is said to be realizable, if it is the spectrum of a nonnegative matrix. In this paper we provide a new sufficient condition for a given list Λ to be universally realizable (UR), that is, realizable for each possible Jordan canonical form allowed by Λ. Furthermore, the resulting matrix (that is explicity provided) is permutative, meaning that each of its rows is a permutation of the first row. In particular, we show that a real Suleĭmanova spectrum, that is, a list of real numbers having exactly one positive element, is UR by a permutative matrix.
{"title":"Permutative universal realizability","authors":"R. Soto, Ana Julio, Jaime H. Alfaro","doi":"10.1515/spma-2020-0123","DOIUrl":"https://doi.org/10.1515/spma-2020-0123","url":null,"abstract":"Abstract A list of complex numbers Λ is said to be realizable, if it is the spectrum of a nonnegative matrix. In this paper we provide a new sufficient condition for a given list Λ to be universally realizable (UR), that is, realizable for each possible Jordan canonical form allowed by Λ. Furthermore, the resulting matrix (that is explicity provided) is permutative, meaning that each of its rows is a permutation of the first row. In particular, we show that a real Suleĭmanova spectrum, that is, a list of real numbers having exactly one positive element, is UR by a permutative matrix.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"66 - 77"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0123","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43265514","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A proof of the statement per(A ∘ B) ≤ per(A)per(B) is given for 4 × 4 positive semidefinite real matrices. The proof uses only elementary linear algebra and a rather lengthy series of simple inequalities.
{"title":"An elementary proof of Chollet’s permanent conjecture for 4 × 4 real matrices","authors":"G. Hutchinson","doi":"10.1515/spma-2020-0126","DOIUrl":"https://doi.org/10.1515/spma-2020-0126","url":null,"abstract":"Abstract A proof of the statement per(A ∘ B) ≤ per(A)per(B) is given for 4 × 4 positive semidefinite real matrices. The proof uses only elementary linear algebra and a rather lengthy series of simple inequalities.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"83 - 102"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0126","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47287352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We combine matrix theory and graph theory methods to give a complete characterization of the surjective linear transformations of tropical matrices that preserve the cyclicity index. We show that there are non-surjective linear transformations that preserve the cyclicity index and we leave it open to characterize those.
{"title":"Linear transformations of tropical matrices preserving the cyclicity index","authors":"A. Guterman, E. Kreines, C. Thomassen","doi":"10.1515/spma-2020-0128","DOIUrl":"https://doi.org/10.1515/spma-2020-0128","url":null,"abstract":"Abstract We combine matrix theory and graph theory methods to give a complete characterization of the surjective linear transformations of tropical matrices that preserve the cyclicity index. We show that there are non-surjective linear transformations that preserve the cyclicity index and we leave it open to characterize those.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"112 - 118"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0128","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47536458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Emre Kişi, M. Sarduvan, H. Özdemir, Nurgül Kalaycı
Abstract We propose an algorithm, which is based on the method given by Kişi and Özdemir in [Math Commun, 23 (2018) 61], to handle the problem of when a linear combination matrix X=∑i=1mciXiX = sumnolimits_{i = 1}^m {{c_i}{X_i}} is a matrix such that its spectrum is a subset of a particular set, where ci, i = 1, 2, ..., m, are nonzero scalars and Xi, i = 1, 2, ..., m, are mutually commuting diagonalizable matrices. Besides, Mathematica implementation codes of the algorithm are also provided. The problems of characterizing all situations in which a linear combination of some special matrices, e.g. the matrices that coincide with some of their powers, is also a special matrix can easily be solved via the algorithm by choosing of the spectra of the matrices X and Xi, i = 1, 2, ..., m, as subsets of some particular sets. Nine of the open problems in the literature are solved by utilizing the algorithm. The results of the four of them, i.e. cubicity of linear combinations of two commuting cubic matrices, quadripotency of linear combinations of two commuting quadripotent matrices, tripotency of linear combinations of three mutually commuting tripotent matrices, and tripotency of linear combinations of four mutually commuting involutive matrices, are presented explicitly in this work. Due to the length of their presentations, the results of the five of them, i.e. quadraticity of linear combinations of three or four mutually commuting quadratic matrices, cubicity of linear combinations of three mutually commuting cubic matrices, quadripotency of linear combinations of three mutually commuting quadripotent matrices, and tripotency of linear combinations of four mutually commuting tripotent matrices, are given as program outputs only. The results obtained are extensions and/or generalizations of some of the results in the literature.
{"title":"On the spectrum of linear combinations of finitely many diagonalizable matrices that mutually commute","authors":"Emre Kişi, M. Sarduvan, H. Özdemir, Nurgül Kalaycı","doi":"10.1515/spma-2020-0138","DOIUrl":"https://doi.org/10.1515/spma-2020-0138","url":null,"abstract":"Abstract We propose an algorithm, which is based on the method given by Kişi and Özdemir in [Math Commun, 23 (2018) 61], to handle the problem of when a linear combination matrix X=∑i=1mciXiX = sumnolimits_{i = 1}^m {{c_i}{X_i}} is a matrix such that its spectrum is a subset of a particular set, where ci, i = 1, 2, ..., m, are nonzero scalars and Xi, i = 1, 2, ..., m, are mutually commuting diagonalizable matrices. Besides, Mathematica implementation codes of the algorithm are also provided. The problems of characterizing all situations in which a linear combination of some special matrices, e.g. the matrices that coincide with some of their powers, is also a special matrix can easily be solved via the algorithm by choosing of the spectra of the matrices X and Xi, i = 1, 2, ..., m, as subsets of some particular sets. Nine of the open problems in the literature are solved by utilizing the algorithm. The results of the four of them, i.e. cubicity of linear combinations of two commuting cubic matrices, quadripotency of linear combinations of two commuting quadripotent matrices, tripotency of linear combinations of three mutually commuting tripotent matrices, and tripotency of linear combinations of four mutually commuting involutive matrices, are presented explicitly in this work. Due to the length of their presentations, the results of the five of them, i.e. quadraticity of linear combinations of three or four mutually commuting quadratic matrices, cubicity of linear combinations of three mutually commuting cubic matrices, quadripotency of linear combinations of three mutually commuting quadripotent matrices, and tripotency of linear combinations of four mutually commuting tripotent matrices, are given as program outputs only. The results obtained are extensions and/or generalizations of some of the results in the literature.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"305 - 320"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0138","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44191201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Isaac Cinzori, Charles R. Johnson, Hannah Lang, Carlos M. Saiago
Abstract Using the recent geometric Parter-Wiener, etc. theorem and related results, it is shown that much of the multiplicity theory developed for real symmetric matrices associated with paths and generalized stars remains valid for combinatorially symmetric matrices over a field. A characterization of generalized stars in the case of combinatorially symmetric matrices is given.
{"title":"Further generalization of symmetric multiplicity theory to the geometric case over a field","authors":"Isaac Cinzori, Charles R. Johnson, Hannah Lang, Carlos M. Saiago","doi":"10.1515/spma-2020-0119","DOIUrl":"https://doi.org/10.1515/spma-2020-0119","url":null,"abstract":"Abstract Using the recent geometric Parter-Wiener, etc. theorem and related results, it is shown that much of the multiplicity theory developed for real symmetric matrices associated with paths and generalized stars remains valid for combinatorially symmetric matrices over a field. A characterization of generalized stars in the case of combinatorially symmetric matrices is given.","PeriodicalId":43276,"journal":{"name":"Special Matrices","volume":"9 1","pages":"31 - 35"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/spma-2020-0119","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43262849","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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